Econ 172A Fall 2004 Final Examination I Instructions 1 The examination has five questions Answer them all 2 If you do not know how to interpret a question then ask me 3 You must justify your answers to each question 4 The table below indicates how points will be allocated on the exam 5 Work alone You may not use notes books or calculators 6 You have three hours 7 If you sign the Buckley waiver attached to the exam then you will be able to pick up your exam in a public area in Sequoyah 245 when the tests are graded on or before December 15 2004 If you do not sign the Buckley waiver then you will be able to pick up your exam from the department undergraduate coordinator beginning January 18 2005 I will accept requests to reconsider grades only from students who sign the Buckley waiver and submit a request in writing after examining but not removing their examination from Sequoyah Hall I try to be especially careful when evaluating borderline cases and have changed about five final grades in the last 26 years Score Possible 50 40 40 60 60 250 500 I II III IV V Exam Total Course Total Grade in Course 1 1 My son Ben is starting a rock band Lit Fuse Ben will do the vocals and play lead guitar Five of his friends will be in the band They have different talents Ben evaluated the relative abilities of his friends and wants to come up with an assignment that maximizes the total quality of the band The table below gives the benefit of assigning a particular boy to a particular instrument So for example if Adam plays Bass Alex plays Drums Chase plays guitar Isaac plays Keyboard and Ryan is the Roadie then the quality if the band is 10 13 18 12 10 63 Adam Alex Chase Isaac Ryan Bass Drum 10 12 17 13 15 5 14 6 14 6 Guitar Keyboard Roadie 6 8 5 10 16 2 18 11 6 16 12 4 16 12 10 a Assume that each boy plays exactly one role instrument or roadie and each role is assigned to exactly one boy Find an assignment of boys to instruments that maximizes the total expected profit Your answer should describe which boy should be assigned to which role and the associated total benefit You must explain how you arrived at the answer and why it solves the problem Properly using the algorithm presented in class with short explanations of the steps is sufficient If you do not use the algorithm you must provided complete and detailed arguments to justify your answer Please note that the objective is to maximize total benefit b Suppose that Chase gets a drum set for Christmas and therefore becomes the band s drummer What is the optimal assignment of roles for the rest of the band Again your answer should describe which boy should be assigned to which role and the associated total benefit You must explain how you arrived at the answer and why it solves the problem 2 2 A fertilizer company has decided to manufacture a large supply of various plant foods to be sold during the upcoming planting season The company can invest up to 25 000 in the three basic ingredients nitrates which cost 800 per ton phosphates which cost 400 per ton and potash which costs 1000 per ton Three standard grades of plant food will be produced from these ingredients regular in which nitrates phosphates and potash are combined respectively in a 3 6 1 ratio by weight extra is a 4 4 1 mixture super is a 6 4 3 mixture Regular can be sold for 750 per ton Extra can be sold for 800 per ton Super can be sold for 900 per ton The company s objective is to maximize profits total sales minus total expenditures for ingredients Its production capacity permits it to manufacture no more than 40 tons of plant food overall a Formulate an LP that would determine how much of each ingredient it should buy and how much of each grade of plant food it should produce b Repeat part a subject to the additional condition that the firm can somehow earn an immediate 10 on all capital not invested in nitrates phosphates and potash Hence the firm can earn 1 on each 10 not spent on the three ingredients Both of your answers must include a definition of the variables in words The definition must include appropriate units for the variables You must also specify the objective function and explain why the function you write down is appropriate and all constraints along with a description of how they correspond to the problem description 3 3 For what values of A is x1 x2 x3 x4 8 0 0 3 a solution to the linear programming problem max subject to Ax1 x1 x1 4x2 4x2 2x2 6x3 8x3 4x3 5x4 2x4 2 3x4 1 x 0 You may use any method to answer this question but you must explain the method that you use and why it works 4 4 Jack and Jill live together Jack works at home and likes to smoke Jill who must spend much of the day away from home hates it when Jack smokes Imagine that each day Jack can do one of four things he can abstain from smoking he can smoke at 1 PM he can smoke at 2 PM or he can smoke at 3 PM he never smokes more than once in a day Jill can check up on Jack exactly once at 1 2 or 3 When Jill checks she can detect signs of smoking in the previous hour so if Jill checks at 2 PM she ll know if Jack smoked at 1 or at 2 Assume that Jack and Jill choose their strategies simultaneously If Jack does not smoke he receives payoff 0 If he smokes and Jill does not find out he receives payoff 1 If he smokes and Jill catches him he receives payoff 1 The game is zero sum Answer the questions on the next page Your answers should provide sufficient justification that it is clear you understand the relevant concepts dominance value and so on 5 a Write down a payoff matrix for this game Label the strategies and explain what they represent b Does Jack have any dominated strategies If so identify the dominated strategies and explain why they are dominated c Does Jill have any dominated strategies If so identify the dominated strategies and explain why they are dominated d Find the pure strategy security levels of both players e Does the game have an equilibrium in pure strategies If so find it f Now assume Jill s checking strategy is constrained in the following way the probability that she checks at 1 PM must be equal to the probability that she checks at 2 PM Write down a payoff matrix for this game To do this assume that Jill has two pure strategies check at 3 PM and check at 1 PM with …